OMM 2011

$25$ lightbulbs are distributed in the following way: the first $24$ are placed on a circumference, placing a bulb at each vertex of a regular $24$ -gon, and the remaining bulb is placed on the center of said circumference. At any time, the following operations may be applied:

• Take two vertices on the circumference with an odd amount of vertices between them, and change the state of the bulbs on those vertices and the center bulb.
• Take three vertices on the circumference that form an equilateral triangle, change the state of the bulbs on those vertices and the center bulb.

Prove from any starting configuration of on and off lightbulbs, it is always possible to reach a configuration where all the bulbs are on.

Let $ABC$ be an acute triangle and $\Gamma$ its circumcircle. Let $l$ be the line tangent to $\Gamma$ at $A$ . Let $D$ and $E$ be the intersections of the circumference with center $B$ and radius $AB$ with lines $l$ and $AC$ , respectively. Prove the orthocenter of $ABC$ lies on line $DE$ .

Let $n$ be a positive integer. Find all real solutions $(a_1, a_2, \dots, a_n)$ to the system:

$$a_1^2 + a_1 - 1 = a_2$$ $$a_2^2 + a_2 - 1 = a_3$$ $$\hspace*{3.3em} \vdots$$ $$a_{n}^2 + a_n - 1 = a_1$$

Find the smallest positive integer that uses exactly two different digits when written in decimal notation and is divisible by all the numbers from $1$ to $9$ .

A $(2^n - 1) \times (2^n +1)$ board is to be divided into rectangles with sides parallel to the sides of the board and integer side lengths such that the area of each rectangle is a power of 2. Find the minimum number of rectangles that the board may be divided into.

Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be two circumferences intersecting at points $A$ and $B$ . Let $C$ be a point on line $AB$ such that $B$ lies between $A$ and $C$ . Let $P$ and $Q$ be points on $\mathcal{C}_1$ and $\mathcal{C}_2$ respectively such that $CP$ and $CQ$ are tangent to $\mathcal{C}_1$ and $\mathcal{C}_2$ respectively, $P$ is not inside $\mathcal{C}_2$ and $Q$ is not inside $\mathcal{C}_1$ . Line $PQ$ cuts $\mathcal{C}_1$ at $R$ and $\mathcal{C}_2$ at $S$ , both points different from $P$ , $Q$ and $B$ . Suppose $CR$ cuts $\mathcal{C}_1$ again at $X$ and $CS$ cuts $\mathcal{C}_2$ again at $Y$ . Let $Z$ be a point on line $XY$ . Prove $SZ$ is parallel to $QX$ if and only if $PZ$ is parallel to $RX$ .